Question

Passing through (-2,1 ) and perpendicular to
4x + 7y + 3 = 0.

Answer 1

7x - 4y + 18 = 0

Explanation:

The slope-intercept form of an equation of a line:

`y=mx+b`

m - slope

b - y-intercept

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Let

`k:y=m_1x+b_1\\\\l:y=m_2x+b_2\\\\l\ \perp\ k\iff m_1m_2=-1\to m_2=-(1)/(m_1)\\\\l\ \parallel\ k\iff m_1=m_2`

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We have the equation of a line in a general form (Ax + By + C = 0)

Convert it to the slope-intercept form:

`4x+7y+3=0` subtract 7y from both sides

`4x+3=-7y` divide both sides by (-7)

`-(4)/(7)x-(3)/(7)=y\to m_1=-(4)/(7)`

Therefore

`m_2=-(1)/(-(4)/(7))=(7)/(4)`

We have the equation:

`y=(7)/(4)x+b`

Put the coordinates of the point (-2, 1) to the equation, and solve for b :

`1=(7)/(4)(-2)+b`

`1=-(7)/(2)+b` multiply both sides by 2

`2=-7+2b` add 7 to both sides

`9=2b` divide both sides by 2

[te]x\dfrac{9}{2}=b\to b=\dfrac{9}{2}[/tex]

Finally:

`y=(7)/(4)x+(9)/(2)` - slope-intercept form

Convert to the general form:

`y=(7)/(4)x+(9)/(2)` multiply both sides by 4

`4y=7x+18` subtract 4y from both sides

`0=7x-4y+18`